Derivative of \( \displaystyle 2 x \left(2 x - 1\right) \)
Problem 2.1341 · medium
Differentiate \( \displaystyle f(x) = 2 x \left(2 x - 1\right) \).
- \[ \frac{d}{d x} 2 x \left(2 x - 1\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(4 x^{2} - 2 x\right) \]algebra algebraExpand the expression using the distributive property. Simplify the coefficients.✓ Proved
- \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 4 x^{2} \]sumApply the difference rule for derivatives.✓ Proved
- \[ = - 2 \frac{d}{d x} x + 4 \frac{d}{d x} x^{2} \]constant-multipleFactor out the constants from each term.✓ Proved
- \[ = 8 x - 2 \]power algebraApply the power rule to each term. Multiply the terms to get the final result.✓ Proved
Answer \( 8 x - 2 \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (error) — Step 1 incorrectly applies the derivative rule directly to the product without using the product rule. Subsequent steps expand the expression inside the derivative, which is not a valid single-rule application and violates the one‑change rule. The solution misrepresents the differentiation process.qwen3.6:27b-mlx: pass — The solution correctly applies one rule per step, uses valid labels from the fixed vocabulary, and provides accurate notes. The expansion and differentiation steps are logically sound.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies one rule per step, uses valid labels from the fixed vocabulary, and provides accurate notes. The expansion and differentiation steps are logically sound.gpt-oss:20b: fail (error) 2026-10-03 — Step 1 incorrectly applies the derivative rule directly to the product without using the product rule. Subsequent steps expand the expression inside the derivative, which is not a valid single-rule application and violates the one‑change rule. The solution misrepresents the differentiation process.qwen3.6:27b-mlx: pass 2026-09-30gpt-oss:20b: inconclusive 2026-09-30 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"Step 1 incorrectly applies the derivative rule to the whole product without using the product rule. The derivative of \(2x(2x-1)\) cannot be taken directl
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-10-03 with SymPy 1.14.0.